Posts tagged: papers
A Penrose tiling can be constructed using just two different tiles in the shape of a thick and thin rhombus:
Here, the angles are multiples of ø = π/5. The edges of these tiles are marked with two different kinds of arrows. This is done to enforce a specific matching rule that ensures that the tiling is non-periodic, which is one of the defining features of Penrose tilings. Different tiles can only be placed next to each other if the touching edges have the same type of arrow and point in the same direction.
If these matching conditions were not in place, then it would be possible to construct tilings which are periodic as shown in the following image. There are translational symmetries present in 8 different directions here:
The image at the top shows a part of a Penrose tiling obeying the matching conditions with the arrows displayed. The numbers are just there to index some other property of directionality not discussed here.
If these matching conditions are satisfied for a complete tiling, then the resulting configuration will always be non-periodic. However, following the matching rules alone does not guarantee an infinite tiling of the entire plane. It is therefore possible to construct finite regions that obey the matching rules, but cannot be extended any further without allowing for periodicities or contradicting the matching rules.
There do exist sets of tiles that will always admit non-periodic tilings in which no extra matching conditions need to be imposed. For a list of such tiles that tile the plane, 3-dimensional space, and even the hyperbolic plane, see this list of aperiodic sets of tiles.
Image sources:
(click through the images to view in high-res)
Penrose tilings are an example of the non-periodic tilings discussed in the last post. Recall that these are tilings that cover the entire infinite plane leaving neither gaps nor overlaps. Whats nice about these tilings is that the set of tiles used to construct the Penrose tilings only consists of two different basic shapes consisting of quadrilaterals.
Whats even more remarkable about Penrose tilings is that using just these two shapes it is possible to construct infinitely many different tilings that cover the infinite plane. This infinity is infinitely bigger in size than the countable infinity of the whole numbers ( 1, 2, 3, 4, and so on), but rather equivalent to the uncountable infinity associated to the real numbers (which includes all whole numbers, fractions, and decimals with infinitely many digits).
Despite the existence of these infinitely many different Penrose tilings, there is one peculiar property they all exhibit. Consider any finite region, or patch, of one particular Penrose tiling. Then it is possible to find an exact copy of this patch in any other different Penrose tiling! Moreover, this patch occurs infinitely often in different spots in any given tiling! Note that this is true of any patch of any size no matter how big as long as its finite. This implies that if you were only able to examine a finite part of any Penrose tiling, you could never really distinguish that entire tiling from any other tiling. Thus, different Penrose tilings are only perfectly distinguishable in the infinite limit of the entire plane.
The images shown above display finite regions of Penrose tilings. They are constructed using an elegant “cut-and-project” method (also used here), which involves projecting the points of the integer lattice in 5-dimensional Euclidean space, onto a certain 2-dimensional plane. Connecting adjacent projected points in this plane by lines then yields a Penrose tiling.
Further reading:
- Penrose Tiles to Trapdoor Ciphers by Martin Gardner
- Algebraic Theory of Penrose’s Non-periodic Tiligns of the Plane by N.G. de Bruijn
- The Empire Problem in Penrose Tilings by Laura Effinger-Dean
Image Source: Wikipedia
For any orthogonal maze shape you can possibly think of, there exists a crease pattern on a flat piece of paper which can be folded into that maze shape. There also exists an algorithm generating the crease pattern given the maze shape that is easy to implement in terms of its algorithmic complexity. You can design your own here.
Shown above is a crease pattern that folds into the “maze” on the left, which is a representation of the letters on the right. Red lines are mountain folds and blue lines are valley folds. I personally just like looking at the crease patterns.
It was not random. The only choice made by me was to follow the models as they are originally described in the papers cited in the post ( [1], [2]. [3]). Let me elaborate more on how the simplified transformation used in the post differs from the one given in the papers, and also on how specific positions of neurons in V1 play a role in the creation and transformation of neural patterns to visual hallucinations. Keep in mind that I am no neuroscientist.
Cowan and Ermentrout in [1] were amongst the first to do such a modeling. The construction of the formula for the coordinate transformation does indeed seem to be motivated by the actual architecture of V1. They state in the paper: “A variety of experimental observations, anatomical, physiological, and psychophysical, have established in primates, and presumably also in humans, that there is a conformal projection of the visual field, onto the visual cortex.”
The formula for the transformation that was used in the post is a simplified version of the actual formula as it appears in the papers in two ways. The complete formula (first appearing on page 2 of [1]) contains a few numerical constants as multiplicative factors, which represent physical parameters of the retina for things like the sizes and numbers of ganglion cells in the eye. So in this regard, the coordinate transformation isn’t necessarily constructed in terms of the “positions of neurons in V1”, but rather in terms of properties of the retina. The first way, the transformation used in the post differs from the one in the paper is through the omission of these physical constants. This is reasonable for the purposes of illustration because the constants mainly serve to define the relevant size scale, and the overall pictures resulting form the transformation remain essentially the same without them.
Also, the original formula for the coordinate transformation has two limiting cases in which the formula can be approximated by simpler ones. One case is for regions that are close to the center of the retina where the transformation turns to the regular transformation from polar coordinates to Cartesian coordinates. The other limiting case occurs for regions that are sufficiently far from the center of the retina. In this case, the transformation becomes the one used in the post (log-polar coordinates).
The authors in [2], which includes Cowan, generalize and extend the original work done in [1]. Its worth noting that [2] was published about 20 years after [1], which was published in 1979. Therefore, since the field of neuroscience has had some time to grow, [2] is able to offer some deeper insights on the matter. The work done in [2] does take into account the relative positions of neurons in V1.
In the papers, and especially in the post, V1 is simply modeled as a flat two dimensional plane. However, in reality V1 possesses a higher dimensional structure consisting of crinkles and folds like the rest of the brain. V1 does more than simply respond to light/dark regions in the visual field with high/low neural activity as described in the post. In addition, neurons on V1 are responsible for detecting contours and edges present in the visual field—that is, the finer boundaries between light and dark regions. Apparently, the depth to V1 consists of collections of neurons referred to as hypercolumns, and it is the neurons in a given hypercolumn which detect the contours in the visual field. The way in which the neurons do this contour detecting is pretty involved in itself and is a primary focus of [2]. Basically, certain neurons in a hypercolum are there to detect only one particular angle that a contour can have, and each hypercolum contains enough different neurons so that contours of all possible angles can be detected. So for each point in the visual field, there exists a hypercolumn in V1 that allows for the proper detection of an edge at any angle in a way that allows for the continuous variation of contours in the visual field.
In the model, hallucinations arise when certain patterns of neural activity are present in V1. The existence of these hypercolumns, and the way neurons are able to interact with neurons in the same and in other hypercolumns, helps explain why certain patterns of neural activity come about (mainly the doubly periodic patterns like the lattice symmetries). It seems that neurons in a given hypercolumn can interact with one another freely, but neurons in different hypercolumns can only interact with each other if the interacting neurons are ones that detect the same contour angle in the visual field. The dynamics of this kind of interaction in V1 is mathematically modeled in [2] and is the main extension that [2] offers to [1]. It is described with more rigor in the paper than I am able to give here. To be honest, I haven’t read it all and am yet to understand all the details myself.
Whether or not you know about or care to use the mathematical coordinate systems described in the last post, it seems that our eyes and brains already make use of them when seeing and hallucinating. How this is the case will be explained here in some detail. Of course, our senses are intricately correlated with the dynamics of our brains in ways so complex that we can barely understand or even comprehend them at all, but there are still ways to reason and model (however crude they may be) the workings of such phenomenon.
Visual hallucinations are a universal human experience which occur in varying degrees in different circumstances—from when we rub or apply pressure to our eyes to more extreme instances like when we are under the influence of psychoactive drugs. What we “see” when we hallucinate is ultimately a unique subjective experience, but there does seem to be some objective similarities amongst reported hallucinations. Some of these hallucinations appear to be fixed in the visual field and do not change as we look around. Moreover, they can even be experienced when the eyes are closed, and have also been reported by people who are blind. This gives reason to believe that the source or cause of these hallucinations may be independent of what we really see and receive as sensory input from the external world, and instead originate on a deeper level within the brain itself.
Studies suggest that a certain region of the brain, the visual cortex (also known as V1), plays a primary role in the processing of visual information. There is a correspondence from what we see in the visual field with our eyes to the neural activity in parts of V1. Spatial input of light and dark on the eyes is translated to “light/dark” regions of high/low neural activity in V1. Basically, images form certain patterns on the retinas of our eyes which are then converted to related patterns on the visual cortex in our brains leading to the perception of the image.
To describe more rigorously the nature of this image translation from the eyes to the brain, we can use mathematics and coordinate systems to define a coordinate mapping between the visual field to the V1 region of the brain. Such a transformation was modeled by J. D. Cowan and G. B, Ermentrout in their 1979 paper “A Mathematical theory of Visual Hallucinations”. A simplified version capturing the essence of their model is described in what fallows.
Interpreting the visual field as a two-dimensional plane, let’s use polar coordinates to label points in the visual field. The center of the visual field is taken as the origin of the coordinate system so that a point P in the visual field is described by a pair of numbers P = ( r , a ), where r is the distance of the point P to the center of the visual field and a is the relative angle the point makes with respect to some fixed axis.
Now we need a way a describe how points in the visual field correspond to points in the V1 region of the brain, but first we need a way to label points in the V1 region. To do this, we model the V1 region as another two-dimensional flat plane. This time, let’s use the normal Cartesian coordinate system to label the points in V1 as pairs of numbers ( x , y ).
The coordinate mapping constructed here will tell us how a point in the visual field, labelled as P = ( r , a ) in polar coordinates, gets translated to the corresponding point in the V1 region of the brain labelled as ( x , y ) in Cartesian coordinates. This ( x , y ) point in V1 is given in terms of the r and a coordinates of the visual field through the relationship
( x , y ) = ( ln r , a ),
where ln r is the natural logarithm of r.
Thus, the point that is given by the polar coordinates ( r , a ) in the visual field is mapped to the corresponding point in V1 whose Cartesian coordinate has a horizontal component of ln r and a vertical component of a. This transformation can be interpreted as log-polar coordinates, and may be recognized as the complex logarithm. We can now use this coordinate transformation to describe the shapes of certain hallucinations.
Normally, we actually receive sensory input through the visual field and corresponding neural patterns are triggered in our brains which result in the perception of whatever we are seeing. The model explained here explains the existence of hallucinations by an opposite mechanism. Without even receiving real sensory input through the visual field, V1 can still be a brain region with high neural activity (perhaps more so when on certain drugs). These patterns of neural activity in V1 may be perceived as if one is actually seeing a pattern in the visual field, but they are really just hallucinations. In the 1920s, psychologist Heinrich Klüver researched himself and others while having ingested mescaline (more info here) in the form of peyote buttons and attempted to classify the visual hallucinations they experienced. The observed hallucinations manifesting themselves as geometrical patterns were classified into four types and were referred to as form constants: 1) tunnels, 2) spirals, 3) cobwebs, and 4) lattices.
The coordinate transformation just described between the visual field and the V1 region of the visual cortex is successful in explaining the occurrence of the form constants.
1) Vertical or horizontal stripes stripes of neural activity on the V1 region may look something like this:
Applying the coordinate transformation in the reverse direction, we see that vertical stripes get mapped to circles in the visual field and horizontal stripes get mapped to rays emanating out from the center of the visual field:
2) Stripes of neural activity in V1 in arbitrary directions such as these diagonals
get mapped to spirals in the visual field:
3) Combinations of stripes such as these vertical and horizontal ones
would then get mapped to patterns that resemble cobwebs:
4) Neural activity in V1 is not limited to stripes. There may be activity with certain lattice symmetries like these checkerboard or honeycomb patterns
which would get mapped to hallucinations that appear like these
This model and these examples are idealized cases, but serve to approximate what may be happening in our brains when certain hallucinations occur. Despite having empirical applications in this context, the coordinate transformation described here can be freely applied to any image resulting in transformed images that still manage to express a psychedelic aesthetic (see here for some examples).
If you ever wondered when or why you would use polar coordinates I hope this post serves as a justifiable application. Conversely, I do not necessarily justify the use of psychedelics to help understand what you are learning in your math lessons.
This post was inspired by these papers, which explain what was attempted here and more in much greater detail:
- “A Mathematical Theory of Visual Hallucinations”
- “Geometric Visual Hallucinations, Euclidean Symmetry and the Functional Architecture of Striate Cortex”
- “A Model for the Origin and Properties of Flicker-Induced Geometric Phosphenes”
Get the Mathematica code for these animations as a CDF file here.
Making use of music theory, group theory, and category theory
From Musical Actions of Dihedral Groups
Abstract:
The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor chords. We illustrate both geometrically and algebraically how these two actions are {\it dual}. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.
Summary:
This paper connects the twelve musical tones to elements in the dihedral group of order 24 (the symmetries of a regular dodecagon). The translation from pitch classes to integers modulo 12 allows for the modeling of musical works using abstract algebra. The first action on major and minor chords described in the paper is based on the musical techniques of transposition and inversion. A transposition moves a sequence of pitches up or down and an inversion reflects a melody about a fixed axis. The other action arises from the P, L, and R operations of the 19th-century music theorist Hugo Riemann. It is through these operations that the dihedral group of order 24 acts on the set of major and minor triads. The paper also describes how the P, L, and R operations have beautiful geometric presentations in terms of graphs. In particular the authors describe a connection between the PLR-group and chord progressions in Beethoven’s 9th Symphony, which leads to a proof that the PLR-group is dihedral. Another musical example is Pachelbel’s Canon in D. In summary, the paper gives a very pretty explanation of what we commonly hear in tonal music in terms of elementary group theory.
Hyperbolic geometry is an example of a non-Euclidean geometry.
Spacetime is assumed to be a branched 4-dimensional manifold embedded in a 6-dimensional Minkowski space. The branches allow quantum interference; each individual branch is a history in the sum-over-histories. A n-manifold embedded in a n+2-space can be knotted. The metric on the spacetime manifold is inherited from the Minkowski space and only allows a particular variety of knots. We show how those knots correspond to the observed particles with corresponding properties. We derive a 2-term Lagrangian. The Lagrangian combined with the geometry of the manifold produces gravity, electromagnetism, weak force, and strong force.
The abstract, topological properties of mathematical knots are related to the properties of elementary particles in the physical universe.
The interested reader can view the paper as a .pdf here.
(via introqft)
David Deutsch is considered to be the father of quantum computation because in 1985 he published a paper which laid down the theoretical foundations for the field. Here is the abstract:
It is argued that underlying the Church-Turing hypothesis there is an implicit physical assertion. Here, this assertion is presented explicitly as a physical principle: ‘every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means’. Classical physics and the universal Turing machine, because the former is continuous and the latter discrete, do not obey the principle, at least in the strong form above. A class of model computing machines that is the quantum generalization of the class of Turing machines is described, and it is shown that quantum theory and the ‘universal quantum computer’ are compatible with the principle. Computing machines resembling the universal quantum computer could, in principle, be built and would have many remarkable properties not reproducible by any Turing machine. These do not include the computation of non-recursive functions, but they do include ‘quantum parallelism’, a method by which certain probabilistic tasks can be performed faster by a universal quantum computer than by any classical restriction of it. The intuitive explanation of these properties places an intolerable strain on all interpretations of quantum theory other than Everett’s. Some of the numerous connections between the quantum theory of computation and the rest of physics are explored. Quantum complexity theory allows a physically more reasonable definition of the ‘complexity’ or ‘knowledge’ in a physical system than does classical complexity theory.
This paper concludes with the following remark:
To view the Church-Turing hypothesis as a physical principle does not merely make computer science a branch of physics. It also makes part of experimental physics into a branch of computer science.
The existence of a universal quantum computer implies that there exists a program for each physical process. In particular, quantum computers can perform any physical experiment. In some cases this is not useful because the result must be known to write the program. But, for example, when testing quantum theory itself, every experiment is genuinely just the running of a quantum computer program.
Quantum computers raise interesting problems for the design of programming languages, which I shall not go into here. From what I have said, programs exist that would (in order of increasing difficulty) test the Bell inequality, test the linearity of quantum dynamics, and test the Everett interpretation. I leave it to the reader to write them.
The quantum computing model is inherently probabilistic. When an algorithm is successfully run on a quantum computer, the output corresponding to the answer to some problem may not always be correct. Instead, we demand that the algorithm gives the correct solution with a probability greater than 1/2. If this is the case, then simply repeating the algorithm multiple times will eventually yield an answer that will be correct with high probability. This is a just a consequence of the Chernoff bound in probability theory. Chris Lomont gives an example of this in The Hidden Subgroup Problem: Review and Open Problems:
Thus the majority is wrong very rarely. For example, we will make most algorithms succeed with probability 3/4 … Although it sounds like a lot, taking 400 repetitions of the algorithm causes our error to drop below 10-20, at which point it is more likely our computer fails than the algorithm fails. And since the algorithms we are considering are usually exponentially faster than classical ones, there is still a net gain in performance. If we do 1000 runs, our error drops below 10-55, at which point it is probably more likely you’ll get hit by lightning while reading this sentence than the algorithm itself will fail.
This is a hyperbolic tessellation with Schläfli symbol {4, 12} shown in the Poincaré disk model and in the band model. The animation displays a rotation of the hyperbolic plane.
An essay on the foundations of geometry, by Bertrand A. W. Russell
1872-1970.
Here is another excerpt I like:
(via blindblannche)
The perfectly conformal grid.
(.cf Escher)
